On the Halpern Method with Adaptive Anchoring Parameters

Abstract

We establish the convergence of a speed-up version of the Halpern iteration with adaptive anchoring parameters in the general geodesic setting of Hadamard spaces, generalizing a recent result by He, Xu, Dong and Mei from a linear to a nonlinear setting. In particular, our results extend the fast rates of asymptotic regularity obtained by these authors for the first time to a nonlinear setting. Our approach relies on a quantitative study of these previous results in the linear setting, combined with certain optimizations and an elimination of the weak compactness arguments employed crucially in the linear setting, which not only allows for the lift of the result to a nonlinear setting but also streamlines the previous convergence analysis considerably. This work is set in the context of recent developments in proof mining, and as byproduct of our approach, we further obtain quantitative information in the form of highly uniform rates of metastability of low complexity, which are new already in the context of Hilbert spaces.